Ramification in Arithmetic and Geometry

Ramiﬁcation in arithmetic and geometry

Anand Deopurkar April 9, 2018

It may seem coincidental that the idea of ramiﬁcation shows up while studying extensions of number ﬁelds and maps between Riemann surfaces. Is this just overuse of terminology, or is there a connection between the two? It turns out that there is a connection; it can be explained by a shared algebraic structure. The goal of this note is to describe this structure, and to explain how it appears in number theory and geometry.

1 Discrete Valuation Rings

The common algebraic structure goes by the name of discrete valuation rings. Here is the deﬁnition. Denition 1. A discrete valuation ring (DVR) is an integral domain R along with a surjective function v : fracR → Z ∪ {+∞}, called the valuation, which satisﬁes the following properties. 1. v(a) = +∞ if and only if a = 0. 2. v(ab) = v(a) + v(b).

3. v(a + b) ≥ min(v(a),v(b)) with equality if v(a) , v(b). 4. a ∈ R if and only if v(a) ≥ 0. We will shortly see three examples of DVRs—one from arithmetic, one from algebra, and one from geometry. The valuation v is often omitted from the notation. This is harmless, because v is often clear from context. In fact, it turns out that there can only be one possible valuation function on a DVR (see Proposition 9). This is only one of the many equivalent deﬁnitions of a DVR—wikipedia lists 10!

1 1.1 DVRs in arithmetic ⊂ Let p be a prime number. Let Zp Q be the set of rational numbers that can be expressed as a/b where a and b are integers and p does not divide b. Note that Zp is a ring, and it contains Z as a sub-ring. In particular, its fraction field is Q. We will shortly see that that R = Zp becomes a DVR with an appropriate valuation v = vp. To define vp, observe that every non-zero rational number r can be written as a r = pn , b ∈ where n,a,b Z and p does not divide a or b. Then we set vp(r) = n. We also set ∞ vp(0) = + , as required. We must verify that vp is a well-defined function on Q. That is, we must check that if there are two ways of representing r as above, then both lead to the same value of v(r). This is easy to do. More interesting (but still straightforward) is the following.

Proposition 2. The ring Zp along with the valuation vp is a DVR. ⊂ Remark 3. Let K be a number field (a finite extension of Q), and OK K the ring of ⊂ ⊂ integers. Let p OK be a prime ideal. We can define OK,p K along with a valuation vp, which is a DVR, generalizing the example above.

1.2 DVRs in algebra ⊂ Let a be a complex number. Let C[x]a C(x) be the set of rational functions that can be expressed as p(x)/q(x), where p(x) and q(x) are polynomials and q(a) , 0. Note that C[x]a is a ring, and contains the polynomial ring C[x] as a sub-ring. In particular, its fraction ﬁeld is C(x). Let f ∈ C(x) be non-zero. Observe that f can be written as p(x) f = (x − a)n , q(x) ∈ ∈ where n Z, and p(x),q(x) C[x] are such that p(a) , 0 and q(a) , 0. Set va(f ) = n. ∞ Set va(0) = + , as required. It is easy to verify that va is well-deﬁned.

Proposition 4. The ring C[x]a along with the valuation va is a DVR.

1.3 DVRs in geometry For our last example, we need some preparation. Let X be a topological space and x ∈ X. Let F(X,x) be the set of pairs (U,f ), where U ⊂ X is an open subset containing

2 x and f is a function f : U → C. Deﬁne an equivalence relation on F(X,x) by saying ∼ ∩ (U1,f1) (U2,f2) if there exists an open set V containing x and contained in U1 U2 | | such that and f1 V = f2 V . An equivalence class of this relation is called a germ of a function on X at x. Denote by FX,x the set of germs of functions on X at x. Said simply, a germ of a fuction on X at x is a function deﬁned in some open set containing x, with the understanding that two functions are considered the same if they agree on some (possibly smaller) open set containing x. For example, the constant function 1 on R and the characteristic function χ of the interval [−1,1] represent the same germ at x = 0. Strictly speaking, a germ is represented by a pair (U,f ), but the U is often omitted. The set of germs of functions on X at x naturally forms a ring—addition and multiplication come from addition and multiplication of functions. Instead of considering all functions, we may restrict ourselves to continuous functions or smooth functions (if X is a manifold). Suppose U ⊂ X is an open set containing x. It is easy to see that we have an isomorphism FX,x FU,x (1) given by restriction of functions. Likewise, if φ: X → Y is a homeomorphism and y = φ(x), then we have an isomorphism FY ,y FX,x (2) given by f 7→ f ◦ φ. Now let X be a Riemann surface. Let OX,x be the be set of germs of holomorphic functions on X at x. Note that if we take a chart centered at x, namely an open set U ⊂ X containing X and a homeomorphism φ: U → V , where V ⊂ C is an open subset such that φ(x) = 0, then by combining Equation 1 and Equation 2, we get an isomorphism OX,x OC,0.

In particular, OX,x does not depend on X or x. This is not surprising; it is simply a reﬂection of the fact that locally near x, a Riemann surface “looks just like” C does near 0. Note that the isomorphism OX,x OC,0 depends on the choice of a chart at x. The ring OC,0 is easy to identify. Recall that C~z denotes the ring of formal power series in a variable z. Let C~zconv be the subset of C~z consisting of power series with a positive radius of convergence (positive includes +∞). For example, 2 ··· 2 ··· f (z) = 1 + z + z + lies in C~zconv, but 0! + 1!z + 2!z + does not. Observe that ⊂ C~zconv C~z is a sub-ring.

Proposition 5. The ring OC,0 is isomorphic to C~zconv.

3 Let us go back to OX,x and show that OX,x becomes a DVR with an appropriate valuation vx. Let η be a non-zero germ of a holomorphic function on X at x represented by (U,f ). Let n be the order of vanishing of f at x. Set

vx(η) = n. ∞ As usual, set vx(0) = + .

Proposition 6. The ring OX,x along with the valuation vx is a DVR.

Let us understand the valuation explicitly on OC,0 C~zconv. Let g be a power series X n g(z) = anz . n 0 ≥ Then v(g) is the minimum n such that an , 0. Using the isomorphism OX,x C~zconv ∈ given by a chart, we get an explicit description of vx on OX,x. If the image of f OX,x is the power series X n g(z) = anz , n 0 ≥ then vx(f ) is the minimum n such that an , 0.

2 Algebraic properties of DVRs

Let R be a DVR with valuation v. Let m ⊂ R be the set of elements that have positive valuation (including +∞). Proposition 7. The set m is a maximal ideal of R. Every element in R \ m is invertible. Consequently, m is the unique maximal ideal of R. A ring with a unique maximal ideal is called a local ring. Proposition 7 says that a DVR is a local ring. Proposition 8. Let t ∈ R be an element with valuation 1. Then t generates m as an ideal. More generally, if I ⊂ R is any ideal, and t ∈ I is an element with minimum valuation, then t generates I as an ideal. Finally, if I ⊂ R is a non-zero ideal, then I = mn for some n ≥ 0. In particular, Proposition 8 says that R is a Principal Ideal Domain (PID). In fact, it says that the only ideals of R are tnR where t ∈ R is an element of valuation 1. An element in R with valuation 1 is called a uniformizer or local parameter. For example, p is a uniformizer in Zp, and z is a uniformizer in OC,0.

4 Proposition 9. Let t ∈ R be a uniformizer. Every element x ∈ R is of the form

x = utn, where u ∈ R is a unit. Consequently, the valuation function v on fracR is unique.

3 Ramication → Let R and S be DVRs with valuations vR and vS. Let φ: R S be a ring homomorphism. Note that φ induces a map of ﬁelds fracR → fracS, which we also denote by φ. ◦ We can compare the two functions vR and vS φ deﬁned on R. Observe that ◦ → ∪ {∞} vS φ: fracR Z is a function satisfying all the properties of a valuation, except possibly surjectivity. The following proposition says that such a function must be a scaled version of the valuation function. → Proposition 10. Let R be a DVR with valuation v. If v0 : R Z 0 is any function satisfying ≥ v0(ab) = v0(a) + v0(b), then there exists a positive integer d such that

v0(a) = d · v(a) for all a ∈ R.

As a result, we conclude that there exists a d such that ◦ · vS φ(a) = d v(a) for all a ∈ R. We say that this integer d is the multiplicity of φ: R → S.

Example 11. Let φ: X → Y be a non-constant holomorphic map between Riemann surfaces. Let x ∈ X and set y = φ(x). The map φ induces a ring homomorphism # → φ : OY ,y OX,x deﬁned by φ#(f ) = f ◦ φ. Then the multiplicity of φ# is the local multiplicity of φ at x.

5 ⊂ ⊂ Example 12. Let K be a number ﬁeld, OK K its ring of integers, and p OK a ∩ → prime ideal. Let Z p = pZ. The inclusion Z OK induces a map → φ: Zp OK,P.

Then the multiplicity of φ is the power of p in the factorization of p into prime ideals of OK . Suppose R/m is a perfect field. We say that φ: R → S is unramied if its multiplicity is 1. Otherwise, we say that it is ramied and its ramication index is d − 1. → Example 13. Let m be a positive integer. The map φ: OC,0 OC,0 induced by z 7→ zm has multiplicity m. If m ≥ 2, then the map is ramified and its ramification index is m − 1.

Example 14. Let K = Q(i), where i2 + 1 = 0, and let P = (1 + i). Then the map → Z2 OK,P has multiplicity 2. Therefore, it is ramiﬁed and its ramiﬁcation index is 1.